If ${\mathrm{deg}}_{x_i}P<\mid S_i\mid$ for each $i$ and $P$ vanishes on $S_1\times \cdots \times S_n$, then $P=0$

[base] If $n=1$, then $P$ is a univariate polynomial of degree below $\mid S_1\mid$ vanishing at every point of $S_1$. By [F1] a nonzero polynomial of that degree cannot have so many roots, so $P=0$.

[ih] Assume the statement is known for polynomials in $n-1$ variables.

Write $P={\sum }_{j=0}^d{P}_j(x_1,\dots ,x_{n-1})x_n^j$ as in [F2], where $d={\mathrm{deg}}_{x_n}P<\mid S_n\mid$.

Fix $(a_1,\dots ,a_{n-1})\in S_1\times \cdots \times S_{n-1}$. Then the univariate polynomial $Q(y):=P(a_1,\dots ,a_{n-1},y)$ has degree at most $d<\mid S_n\mid$ and vanishes on all of $S_n$, so [F1] gives $Q=0$. Therefore every coefficient $P_j(a_1,\dots ,a_{n-1})$ is $0$.

Since the point $(a_1,\dots ,a_{n-1})$ was arbitrary, every coefficient polynomial $P_j$ vanishes on $S_1\times \cdots \times S_{n-1}$. Its degree in each variable is still below the corresponding $\mid S_i\mid$, so the induction hypothesis gives $P_j=0$ for every $j$. Hence $P=0$.